Let n≥ 1 and let A:R^n→ M_n(C) be a measurable matrix-valued function for which there exist constants lambda>0 and Lambda<∞ such that, for almost every x∈R^n and all xi,eta∈C^n, operatornameRebig langle A(x)xi,xi bigrangle ≥ lambda|xi|^2,qquad big|big langle A(x)xi,eta bigrangle big| ≤ Lambda,|xi|,|eta| . Let L=-operatornamediv(A nabla) be the operator in L^2(R^n) associated with the sesquilinear form J(f,g)=int_R^nbig langle A(x)nabla f(x),nabla g(x)big rangle,dx,qquad f,g∈ H^1(R^n), that is, f∈D(L) with Lf=u if and only if f∈ H^1(R^n), u∈ L^2(R^n) and J(f,g)=langle u,g rangle for all g∈ H^1(R^n). Then L is maximal accretive, so its accretive square root √L=L^1/2 is defined by the holomorphic functional calculus for L, and Dbig(√Lbig)=H^1(R^n), with constants 0<c≤ C<∞ depending only on n,lambda,Lambda such that c,|nabla f|_2 ≤ big|√L,f big|_2 ≤ C,|nabla f|_2qquadfor all f∈ H^1(R^n).

Kato's conjecture

About 73 years old · traced to 1953

Let $n\ge 1$ and let $A:\mathbb{R}^n\to M_n(\mathbb{C})$ be a measurable matrix-valued function for which there exist constants $\lambda>0$ and $\Lambda<\infty$ such that, for almost every $x\in\mathbb{R}^n$ and all $\xi,\eta\in\mathbb{C}^n$ ,

\operatorname{Re}\big\langle A(x)\xi,\xi\big\rangle\ \ge\ \lambda|\xi|^2,\qquad \big|\big\langle A(x)\xi,\eta\big\rangle\big|\ \le\ \Lambda\,|\xi|\,|\eta| .

Let $L=-\operatorname{div}(A\nabla)$ be the operator in $L^2(\mathbb{R}^n)$ associated with the sesquilinear form

J(f,g)=\int_{\mathbb{R}^n}\big\langle A(x)\nabla f(x),\nabla g(x)\big\rangle\,dx,\qquad f,g\in H^1(\mathbb{R}^n),

that is, $f\in\mathcal{D}(L)$ with $Lf=u$ if and only if $f\in H^1(\mathbb{R}^n)$ , $u\in L^2(\mathbb{R}^n)$ and $J(f,g)=\langle u,g\rangle$ for all $g\in H^1(\mathbb{R}^n)$ . Then $L$ is maximal accretive, so its accretive square root $\sqrt{L}=L^{1/2}$ is defined by the holomorphic functional calculus for $L$ , and

\mathcal{D}\big(\sqrt{L}\big)=H^1(\mathbb{R}^n),

with constants $0<c\le C<\infty$ depending only on $n,\lambda,\Lambda$ such that

c\,\|\nabla f\|_{2}\ \le\ \big\|\sqrt{L}\,f\big\|_{2}\ \le\ C\,\|\nabla f\|_{2}\qquad\text{for all } f\in H^1(\mathbb{R}^n).

References

Primary source

Wikipedia

Additional references

Wikipedia, Kato's conjecture, the article this problem comes from.

unsolved-problem

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

The conjecture was proved affirmatively more than two decades ago, and a recent similarly named counterexample concerns a different problem.

Tosio Kato posed the conjecture in 1953 for uniformly elliptic divergence-form operators. It asserts that the square-root domain equals $H^1(\mathbb{R}^n)$ with equivalent gradient and square-root norms; this stated problem is solved.

Known results

Hofmann, Lacey, and McIntosh, 2002: proved the result under Gaussian heat-kernel bounds.
Auscher, Hofmann, Lacey, McIntosh, and Tchamitchian, 2001/2002: proved it in full generality for bounded measurable complex coefficients.
Auscher, Axelsson, and McIntosh, 2010: gave a self-contained quadratic-estimate framework and extensions to higher-order elliptic systems.

August 2026 naming collision

An August 2026 preprint by Frank and Ivanisvili is described as a counterexample to a different conjecture also called Kato’s, involving positivity of operator commutators and predicted strip widths. It does not challenge the divergence-form square-root theorem; the associated posts credit AI generally, without naming a model.

Current status (as of August 2026): The stated Kato square-root conjecture is resolved in full generality; no part of this formulation remains open.

Sources

No solutions have been posted yet.